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14
Complexity and Algorithms for Reasoning About Time: A GraphTheoretic Approach
, 1992
"... Temporal events are regarded here as intervals on a time line. This paper deals with problems in reasoning about such intervals when the precise topological relationship between them is unknown or only partially specified. This work unifies notions of interval algebras in artificial intelligence ..."
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Cited by 90 (11 self)
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Temporal events are regarded here as intervals on a time line. This paper deals with problems in reasoning about such intervals when the precise topological relationship between them is unknown or only partially specified. This work unifies notions of interval algebras in artificial intelligence with those of interval orders and interval graphs in combinatorics. The satisfiability, minimal labeling, all solutions and all realizations problems are considered for temporal (interval) data. Several versions are investigated by restricting the possible interval relationships yielding different complexity results. We show that even when the temporal data comprises of subsets of relations based on intersection and precedence only, the satisfiability question is NPcomplete. On the positive side, we give efficient algorithms for several restrictions of the problem. In the process, the interval graph sandwich problem is introduced, and is shown to be NPcomplete. This problem is als...
Graph Colorings and Related Symmetric Functions: Ideas and Applications
, 1998
"... this paper we will report on further work related to this symmetric function ..."
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Cited by 35 (2 self)
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this paper we will report on further work related to this symmetric function
UNLABELED (2 + 2)FREE POSETS, ASCENT SEQUENCES AND PATTERN AVOIDING PERMUTATIONS
"... Abstract. We present bijections between four classes of combinatorial objects. Two of them, the class of unlabeled (2 + 2)free posets and a certain class of chord diagrams (or involutions), already appear in the literature. The third one is a class of permutations, defined in terms of a new type of ..."
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Cited by 23 (9 self)
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Abstract. We present bijections between four classes of combinatorial objects. Two of them, the class of unlabeled (2 + 2)free posets and a certain class of chord diagrams (or involutions), already appear in the literature. The third one is a class of permutations, defined in terms of a new type of pattern. An attractive property of these patterns is that, like classical patterns, they are closed under the action of D8, the symmetry group of the square. The fourth class is formed by certain integer sequences, called ascent sequences, which have a simple recursive structure and are shown to encode (2 + 2)free posets, chord diagrams and permutations. Our bijections preserve numerous statistics. We also determine the generating function of these classes of objects, thus recovering a series obtained by Zagier for chord diagrams. That this series also counts (2 + 2)free posets seems to be new. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern 3¯152¯4, and enumerate those permutations, thus settling a conjecture of Lara Pudwell. 1.
n! MATCHINGS, n! POSETS
, 2010
"... Abstract. We show that there are n! matchings on 2n points without socalled left (neighbor) nestings. We also define a set of naturally labeled (2+2)free posets and show that there are n! suchposetsonn elements. Our work was inspired by BousquetMélou, Claesson, Dukes and Kitaev [J. Combin. Theory S ..."
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Cited by 5 (1 self)
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Abstract. We show that there are n! matchings on 2n points without socalled left (neighbor) nestings. We also define a set of naturally labeled (2+2)free posets and show that there are n! suchposetsonn elements. Our work was inspired by BousquetMélou, Claesson, Dukes and Kitaev [J. Combin. Theory Ser. A. 117 (2010) 884–909]. They gave bijections between four classes of combinatorial objects: matchings with no neighbor nestings (due to Stoimenow), unlabeled (2 + 2)free posets, permutations avoiding a specific pattern, and socalled ascent sequences. We believe that certain statistics on our matchings and posets could generalize the work of BousquetMélou et al., and we make a conjecture to that effect. We also identify natural subsets of matchings and posets that are equinumerous to the class of unlabeled (2 + 2)free posets. We give bijections that show the equivalence of (neighbor) restrictions on nesting arcs with (neighbor) restrictions on crossing arcs. These bijections are thought to be of independent interest. One of the bijections factors through certain uppertriangular integer matrices that have recently been studied by Dukes and Parviainen [Electron. J. Combin. 17 (2010) #R53]. 1.
Enumerating (2+2)free posets by the number of minimal elements and other statistics
 In 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC
, 2010
"... Abstract. A poset is said to be (2 + 2)free if it does not contain an induced subposet that is isomorphic to 2 + 2, the union of two disjoint 2element chains. In a recent paper, BousquetMélou et al. found, using so called ascent sequences, the generating function for the number of (2 + 2)free po ..."
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Abstract. A poset is said to be (2 + 2)free if it does not contain an induced subposet that is isomorphic to 2 + 2, the union of two disjoint 2element chains. In a recent paper, BousquetMélou et al. found, using so called ascent sequences, the generating function for the number of (2 + 2)free posets: P (t) = ∑ ( i n≥0 1 − (1 − t) ). n i=1 We extend this result by finding the generating function for (2 + 2)free posets when four statistics are taken into account, one of which is the number of minimal elements in a poset. We also show that in a special case when only minimal elements are of interest, our rather involved generating function can be rewritten in the form P (t, z) = n,k≥0 pn,kt n z k = 1 + ∑ n≥0 of size n with k minimal elements. zt (1−zt) n+1 n i=1 (1 − (1 − t)i) where pn,k equals the number of (2 + 2)free posets Résumé. Un poset sera dit (2 + 2)libre s’il ne contient aucun sousposet isomorphe à 2 + 2, l’union disjointe de deux chaînes à deux éléments. Dans un article récent, BousquetMélou et al. ont trouvé, à l’aide de “suites de montées”, la fonction génératrice des nombres de posets (2 + 2)libres: c’est P (t) = ∑ ( i n≥0 1 − (1 − t) ). n i=1 Nous étendons ce résultat en trouvant la fonction génératrice des posets (2 + 2)libres rendant compte de quatre statistiques, dont le nombre d’éléments minimaux du poset. Nous montrons aussi que lorsqu’on ne s’intéresse qu’au nombre d’éléments minimaux, notre fonction génératrice assez compliquée peut être simplifiée en P (t, z) = n,k≥0 pn,kt n z k = 1 + ∑ n≥0 taille n avec k éléments minimaux. zt (1−zt) n+1
Minimizing Transceivers in Optical Path Networks
, 2006
"... We consider the problem of routing traffic on lightpaths in unidirectional, WDM path networks with the goal of minimizing the number of transceivers. We show that the problem is NPhard, even in the special case that all traffic requests are destined for a single egress node. In the case of egress t ..."
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We consider the problem of routing traffic on lightpaths in unidirectional, WDM path networks with the goal of minimizing the number of transceivers. We show that the problem is NPhard, even in the special case that all traffic requests are destined for a single egress node. In the case of egress traffic, we give a simple heuristic that will never be worse than twice the optimal.
Selfdual interval orders and rowFishburn matrices
"... Recently, Jelínek derived that the number of selfdual interval orders of reduced size n is twice the number of rowFishburn matrices of size n by using generating functions. In this paper, we present a bijective proof of this relation by establishing a bijection between two variations of uppertria ..."
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Recently, Jelínek derived that the number of selfdual interval orders of reduced size n is twice the number of rowFishburn matrices of size n by using generating functions. In this paper, we present a bijective proof of this relation by establishing a bijection between two variations of uppertriangular matrices of nonnegative integers. Using the bijection, we provide a combinatorial proof of the refined relations between selfdual Fishburn matrices and rowFishburn matrices in answer to a problem proposed by Jelínek.
A matrix for counting paths in . . .
, 1996
"... We define a matrix A associated with an acyclic digraph 1, such that the coefficient of z j in det(I+zA) is the number of jvertex paths in 1. This result is actually a special case of a more general weighted version. ..."
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We define a matrix A associated with an acyclic digraph 1, such that the coefficient of z j in det(I+zA) is the number of jvertex paths in 1. This result is actually a special case of a more general weighted version.
Counting selfdual interval orders
"... Abstract. In this paper, we first derive an explicit formula for the generating function that counts unlabeled interval orders (a.k.a. (2 + 2)free posets) with respect to several natural statistics, including their size, magnitude, and the number of minimal and maximal elements. In the second part ..."
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Abstract. In this paper, we first derive an explicit formula for the generating function that counts unlabeled interval orders (a.k.a. (2 + 2)free posets) with respect to several natural statistics, including their size, magnitude, and the number of minimal and maximal elements. In the second part of the paper, we derive a generating function for the number of selfdual unlabeled interval orders, with respect to the same statistics. Our method is based on a bijective correspondence between interval orders and uppertriangular matrices in which each row and column has a positive entry. Résumé. Dans cet article, on obtient une expression explicite pour la fonction génératrice du nombre des ensembles partiellement ordonnés (posets) qui évitent le motif (2 + 2). La fonction compte ces ensembles par rapport à plusieurs statistiques naturelles, incluant le nombre d’éléments, le nombre de niveaux, et le nombre d’éléments minimaux et maximaux. Dans la deuxième partie, on obtient une expression similaire pour la fonction génératrice des posets autoduaux évitant le motif (2 + 2). On obtient ces résultats à l’aide d’une bijection entre les posets évitant (2 + 2) et les matrices triangulaires supérieures dont chaque ligne et chaque colonne contient un élément positif.
unknown title
"... Abstract. We show that there are n! matchings on 2n points without, so called, left (neighbor) nestings. We also define a set of naturally labeled (2 + 2)free posets, and show that there are n! such posets on n elements. Our work was inspired by BousquetMélou, Claesson, Dukes and Kitaev [J. Combin ..."
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Abstract. We show that there are n! matchings on 2n points without, so called, left (neighbor) nestings. We also define a set of naturally labeled (2 + 2)free posets, and show that there are n! such posets on n elements. Our work was inspired by BousquetMélou, Claesson, Dukes and Kitaev [J. Combin. Theory Ser. A. 117 (2010) 884–909]. They gave bijections between four classes of combinatorial objects: matchings with no neighbor nestings (due to Stoimenow), unlabeled (2 + 2)free posets, permutations avoiding a specific pattern, and so called ascent sequences. We believe that certain statistics on our matchings and posets could generalize the work of BousquetMélou et al. and we make a conjecture to that effect. We also identify natural subsets of matchings and posets that are equinumerous to the class of unlabeled (2 + 2)free posets. We give bijections that show the equivalence of (neighbor) restrictions on nesting arcs with (neighbor) restrictions on crossing arcs. These bijections are thought to be of independent interest. One of the bijections maps via certain uppertriangular integer matrices that have recently been studied by Dukes and Parviainen [Electron. J. Combin. 17 (2010) #R53] Résumé. Nous montrons qu’il y a n! couplages sur 2n points sans emboîtement (de voisins) à gauche. Nous définissons aussi un ensemble d’EPO (ensembles partiellement ordonnés) sans motif (2 + 2) naturellement étiquetés, et montrons qu’il y a n! tels EPO sur n éléments. Notre travail a été inspiré par BousquetMélou, Claesson, Dukes